OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ADVANCED ARGUMENT

摘要

We consider a differential equation u~((n))(t) + p(t) |u(σ(t))|~(μ(t))signu(σ(t)) = 0. where p∈L_(loc)(R_+: R_+), μ ∈ C(R_+: (0. +∞)), σ ∈ C(R_+; R_+), and σ(t) ≥ t for t ∈ R_+. We say that the equation is almost linear if the condition lim_(t→+∞)μ(t) = 1 is satisfied. At the same time, if limsup_(t→+∞)μ(t) ≠ 1 or liminf_(t→+∞)μ(t) ≠ 1, then Eq. is called an essentially nonlinear differential equation. The oscillatory properties of almost linear differential equations have been extensively studied. In the paper, new sufficient (necessary and sufficient) conditions are established for a general class of essentially nonlinear functional differential equations to have Property A.